Bahadur efficiency of the maximum likelihood estimator and one-step estimator for quasi-arithmetic means of the Cauchy distribution

Abstract

Some quasi-arithmetic means of random variables easily give unbiased strongly consistent closed-form estimators of the joint of the location and scale parameters of the Cauchy distribution. The one-step estimators of those quasi-arithmetic means of the Cauchy distribution are considered. We establish the Bahadur efficiency of the maximum likelihood estimator and the one-step estimators. We also show that the rate of the convergence of the mean-squared errors achieves the Cramer-Rao bound. Our results are also applicable to the circular Cauchy distribution.

23 pages; numerical computations added; to appear in Annals of the Institute of Statistical Mathematics

References in corpus (3)

• A closed-form formula for the Kullback-Leibler divergence between Cauchy distributions
• Characterizations of the maximum likelihood estimator of the Cauchy distribution
• Limit theorems for quasi-arithmetic means of random variables with applications to point estimations for the Cauchy distribution

Cited by in corpus (5)

• Characterizations of the maximum likelihood estimator of the Cauchy distribution
• On $f$-divergences between Cauchy distributions
• Limit theorems for quasi-arithmetic means of random variables with applications to point estimations for the Cauchy distribution
• Properties of complex-valued power means of random variables and their applications
• Confidence disc and square for Cauchy distributions